52 ANALYTICAL MECHANICS Therefore the total frictional force and the total frictional torque are, respectively, F = nal I *p de Jo and G=iM*lf'pde. Jo In order to carry out the integral of the foregoing expressions we have to make some assumption with regard to the nature FIG. 38. of dependence of p upon 6. But whatever the relation between p and 0 it is obvious that the sum, over all the surfaces of contact, of the vertical component of the normal reaction must equal the load which rests upon the bearings. If P denotes this load, then p must satisfy the condition XA p sin 6 • dA = al IP sin 0 dO} JQ where A is the total area of contact. ILLUSTRATIVE EXAMPLE. The normal pressure on the bearings is given by the relation p = p0 sin 6; find the total frictional force and the total frictional torque.